Work, Power, and Energy Lecture 8

Transcription

1 Work, Power, and Energy Lecture 8 ˆ Back to Earth... ˆ We return to a topic touched on previously: the mechanical advantage of simple machines. In this way we will motivate the definitions of work, power, and energy. ˆ This will actually provide us a different way of analyzing the motion of particles. With advantages and disadvantages, this energy framework adds some powerful tools to our toolbox for analyzing the motion of material systems. 1

2 Machines and Mechanical Advantage Lecture 8 ˆ The purpose of any mechanical machine is to multiply the input force in order to create a much larger force for useful work. ˆ This multiplication factor is called the mechanical advantage of the machine. ˆ It is possible to break the analysis of a machine into components each connected together. ˆ These components are called simple machines and are traditionally classified as: Lever Wheel and axle Pulley Inclined plane Wedge Screw ˆ This list could be reduced to two: the lever and the inclined plane. ˆ The first three all operate based on a twisting motion around a pivot, while the second three operate based on splitting the support force that counter-balances a perpendicular force. 2

3 Work, Power, and Efficiency Lecture 8 ˆ As was mentioned in an earlier lecture, every machine gains its mechanical advantage by multiplying the displacement over which the force is to be applied. This implies that the product of the two (force and displacement) is presevered. This quantity is called work. The SI unit for work is the joule. ˆ For any real machine, there are frictional effects that limit its efficiency. Work gives us a way to quantify this efficiency as e = W out /W in ˆ Notice that work is a technical term somewhat different than the English term. For example, holding a box without moving it requires no physical work, though much effort may be necessary. ˆ Also, time is not a factor. Whether the box is moved fast or slow, if the displacement and force is the same, so is the work value. ˆ The term power is the rate at which work is done by a machine. The more powerful machine will perform the same amount of work faster. The SI unit is the watt. One horsepower is 746 watts. ˆ Though both force and displacement are vectors, work is not. In order to account for direction we therefore define work as W = F d cos θwhere : math : θ representstheanglebetweenthetwo. I prefer to associate the cosine factor with the force and say that work is defined as the displacement times the component of the force creating the displacement. Work can be negative if θ > 90. In other words, if the force opposes the displacement, the work done is negative. 3

4 Energy is the Ability to Do Work Lecture 8 ˆ The ability to do work is a valuable quality in any machine. Therefore, for any physical system we define this ability to do work as its energy. ˆ You should think of the energy of a system as a property of the system. Whenever a system does work on another system, this represents a transfer of energy from the one to the other. The energy level of one decreases while the energy level of the other increases. ˆ By definition then, we have E = W on W by where W by represents the work done by the system, therefore draining its energy. W on is the work done on the system by external forces. ˆ This is sometimes called the work-energy theorem, but I hope you see it is simply a translation of the definition of energy. ˆ Notice that we are really starting to talk in terms of a system. One of the advantages of the energy approach is that we will be able to make statements about this system without requiring complete knowledge of how the internal parts interact. ˆ We say that energy is a property of the system and work represents the flow of energy across its boundary into and from the system s surroundings. ˆ Energy is like mechanical currency: the system spends and earns it through work. 4

5 Kinetic Energy Defined Lecture 8 ˆ The simplest system of all is the particle. The simplest force of all is the constant force. If we combine them, what happens? According to Newton s 2nd law, acceleration. ˆ How much work is involved in accelerating a particle up to a particular speed? ˆ This is actually a problem we can solve. If we take the fifth kinematic equation of constant accleration and multiply both sides by the mass m we get mv 2 = 2mad where we use the fact that v 0 = 0. ˆ Substitute Newton s second law into the right side and divide both sides by two: W = F d = 1 2 mv2 ˆ This also represents the work that can be done by the particle (imagine slamming it into the wall). We define this as the kinetic energy of the particle: KE = 1 2 mv2 ˆ Whenever there is motion there is kinetic energy because this motion can be captured and converted into useful work. 5

6 Potential Energy Defined Lecture 8 ˆ Throw a ball in the air. It starts with a certain amount of kinetic energy. As the ball rises, where does this energy go? ˆ We have a couple of options. First, we can say that the weight of the object (or the gravitational force) does negative work on the ball by opposing its displacement. ˆ This work represents the energy lost by the ball. The work done by gravity is simply W = mgh where h is the height of the throw. ˆ But we know that what goes up must come down. In fact, we know that if we ignore air drag the final velocity of the ball is the same as the initial velocity. ˆ In other words, the final kinetic energy is the same as the initial kinetic energy. It looks like the energy simply plays hide-and-seek: as the ball rises the energy hides, as it falls the energy shows up again. ˆ Think also about a fully charged battery, or a wind-up toy. All of these things are systems full of energy, ready to do work even though they are not in motion. ˆ We call this potential energy. ˆ For the ball, we say that it has a certain amount of potential energy by virtue of its height: P E = mgh ˆ For systems with other internal forces, other formulas for potential energy will apply. We will meet more as we proceed through this course. ˆ The potential energy related to Newton s law of gravity is GM m/r. 6

7 Conservation of Energy Lecture 8 ˆ Thinking about energy this way, we are including the source of the force (the earth) into the system and defining the potential energy relative to the internal configuration of the system (the height). ˆ The overall energy is conserved, but the (now) internal forces transform the energy from kinetic to potential and back again. ˆ If a system is isolated from its surroundings, the work-energy theorem tells us that the total energy of the system can not change, but this total may be redistributed. ˆ This is called the conservation of energy. ˆ Since this follows by defintion, it is easy to overlook the importance of this principle. In fact, some have argued that this is the most important insight in physics it is certainly one of the most fruitful. In fact, in quantum mechanics the concept of force loses meaning. Even in relativity, force is a bit difficult to define. But energy remains. ˆ For now we simply use energy to calculate and solve mechanical problems. One immediate practical advantage is that energy is not a vector. Using it only involves simple algebra. ˆ The method is straight-forward. We need to catalogue all types of energy in the problem. For some moment in time we need to determine all these values and add them up. This is the total energy. ˆ If energy is conserved, we know that the energy has this value for every other moment. Shift focus to calculate the values of energy for the moment of interest and solve. One disadvantage to using energy is that we cannot answer questions about time precisely because energy is conserved across time. 7

8 Non-Conservative Forces Lecture 8 ˆ Now the bad news: not all forces conserve energy. Some forces destroy energy any form of friction or air drag will do this. ˆ Therefore, energy changes are due to external forces or internal non-conservative forces: E = W on W by W lost ˆ When we defined potential energy, the critical thing was that the kinetic energy at the beginning was all there at the end. This is why we were able to consider the energy as hiding. ˆ But for non-conservative forces, it is not that way. We do not expect a block sliding across a table to leap back into motion after it is brought to rest by friction. ˆ The critical test is the round trip. If all the work done by the force is released when the system returns to its original state, the force is conservative. This is why the potential energy only depends on the state of the system rather than the means whereby it is rearranged. If the potential energy did depend on the path, a round trip would not necessarily sum to zero work done on one side of the trip could be different than the other side. One could use this fact to extract an unlimited amount of energy by running the trip over and over. ˆ But for a non-conservative force, a round-trip results in a net loss of energy. Since the force of friction always opposes the motion, it only does negative work. 8

9 Energy Diagrams Lecture 8 ˆ An energy diagram effectively summarizes the dynamics of a system. ˆ If the state of the system can be represented by one parameter (e.g., the distance between two parts), we can plot the potential energy against this parameter. ˆ On this chart, the force will always point against the slope of the potential energy, pushing the system into areas of low potential energy. ˆ The total energy of the system can be represented as a horizontal line across the chart. The gap between the two represents the kinetic energy of the system. ˆ Since the kinetic energy is never negative, the system cannot ever be in a state where the potential energy exceeds the total. These areas are called the forbidden regions for the system. ˆ The points of intersection are called turning points. They are called this because in these states, the kinetic energy must be zero. Like a ball rising to the top of its trajectory and falling back to earth, the system will approach these states, touch them, then return. ˆ The bowl shaped areas represent stable equilibrium. The system will oscillate between the turning points around stable equilibrium. ˆ Unstable equilibrium are the hills. The system is pushed away from these states. ˆ Finally, friction will have the effect of pulling down the overall energy represented by the horizontal line. As this line falls, it will force the system to find a region of stable equilibrium. ˆ Eventually, friction will pull the system down to the nearest state of stable equilibrium. Friction is always doing this this is why stable equilibrium is so prevelant in the world. 9